This paper suggests an alternative explanation for the recently documented betting against beta anomaly. Given that the equity of a levered firm is equivalent to a call option on firm assets and option returns are non-linearly related to underlying stock returns, linear CAPM-type regressions are generally misspecified. We derive theoretical expressions for the pricing error and analyze its magnitude using numerical examples. Consistent with the empirical findings of Frazzini and Pedersen (2014), our pricing errors are negative, increase with leverage, and become economically significant for higher levels of firm leverage.

Notable quotations from the academic research paper:

"In this paper, we suggest a possible alternative explanation for the betting against beta phenomenon. We propose that the betting against beta phenomenon is due to pricing errors, which arise given that the CAPM does not take non-linearities in stock returns into account. Our rationale is as follows. As highlighted by the classic Black-Scholes-Merton model of corporate debt and equity valuation, the equity of a levered firm is equivalent to a call option written on the underlying value of the firm’s assets. As is known, option returns are highly skewed and non-linearly related to the returns of the underlying. Therefore, linear CAPM-type regressions of equity returns may suffer from model misspecification. Using the Black-Scholes-Merton model, we derive expressions for the model pricing error under the standard CAPM and analyze its magnitude using numerical examples.

Our analysis highlights that the pricing error is negative and becomes economically large as firm leverage increases. That is, consistent with the empirical findings of Frazzini and Pedersen (2014), our theoretical analysis predicts that a portfolio that is long low-beta stocks and short high-beta stocks generates a positive CAPM alpha. However, since the equity is correctly priced under our Black-Scholes-Merton framework, the observed positive alpha is due to the pricing error that is induced by the inadequate linearity assumption of the CAPM. This result questions whether the betting against beta phenomenon is indeed an asset pricing anomaly or whether it is due to the fact that the standard CAPM is an inappropriate setting for analyzing the equity returns of highly levered firms. As the analysis presented in this paper is purely theoretical, our aim here is not to assert that the documented betting against beta phenomenon can fully be attributed to the pricing error that we point out. Such detailed empirical tests are beyond the scope of the present paper. Nonetheless, our findings highlight that care must be taken when we interpret the negative alphas of high-beta stocks as an asset pricing anomaly"

We use industry multiples-based market-to-book decomposition of Rhodes-Kropf, Robinson and Viswanathan (2005) to study the value premium. The market-to-value component drives all of the value strategy return, while the value-to-book component exhibits no return predictability in both portfolio sorts and firm-level return regressions controlling for other stock characteristics. Prior results in the literature linking value/glamor to expectational errors and limits to arbitrage hold due to the market-to-value component, whereas the results linking market-to-book to cashflow risk, exposure to investment-specific technology shocks, and analyst’s risk ratings hold only for the unpriced value-to-book. Overall, our evidence points towards the mispricing explanation for the value premium.

Notable quotations from the academic research paper:

"In this paper, we shed further light on the origins of the “value premium” using a market-to-book decomposition proposed by Rhodes–Kropf, Robinson, and Viswanathan (2005) (RRV hereafter) in their study of misvaluation and merger waves. In particular, the market-to-book ratio is decomposed into market-to-value and value-to-book components, where value is an estimate of fundamental value based on industry multiples conditional on a set of observable characteristics. The market-to-value component represents deviation of the stock price from fundamental value implied by long-run industry valuations, and the value-to-book component represents the “expected” industry-specific valuation of the firm’s net assets. The market-to-value component can be further decomposed into firm-specific deviation of the stock price from contemporaneous industry-level valuation, and the deviation of industry-level valuation from its long-run average.

We find that all of the return predictability of the market-to-book ratio is concentrated in the market-to-value component. Over the 1975-2013 period, a long-short portfolio strategy based on the conventionally used market-to-book ratio produces an average return of 1.42% per month (17.04% annualized). The same strategy based on the market-to-value component produces an average return of 1.56% (18.72% annualized), while going long low value-to-book and short high value-to-book stocks produces an average return of 0.27% per month – statistically insignificant. The Sharpe ratios of the market-to-value strategies are also superior to that of the conventional value strategy. Further decomposition of the market-to-value component into firm-level deviations from contemporaneous industry-level valuations and industry-level deviations from long-run averages shows that it is the firm-specific component that drives return predictability.

If high (low) market-to-value stocks are over (under)priced, we should find that investors are negatively (positively) surprised by their earnings announcements following portfolio formation. This is exactly what we find. We also find that high (low) market-to-value stocks experience positive (negative) earnings surprises in the quarters prior to portfolio formation, suggesting that the mechanism by which these stocks become mispriced is investor overextrapolation of positive (negative) news, leading to over (under)valuation that gets corrected over subsequent quarters as the true fundamentals are revealed. These patterns are not there for the value-to-book component.

We then examine the role of limits to arbitrage, such as short sale constraints and noise trader risk, in the sustaining of mispricing (De Long et al. (1990), Shleifer and Vishny (1997), Pontiff (2006)). In the absence of limits to arbitrage, overvaluation should not persist for long periods of time as the rational arbitrageurs trade against the mispricing. We find that the performance of the market-to-value strategy is concentrated in portfolios characterized by short sale constraints, as captured by institutional ownership and the existence of exchange-traded options. Moreover, these differences are largely due to high market-to-value stocks – the ones going into the short leg of the strategy – where short sale constraints are really binding. We also find that the market-to-value strategy returns are largely due to stocks characterized by noise trader risk as captured by idiosyncratic return volatility. Again, these effects are not there for value-to-book: the mispricing-based explanations of the market-to-book effect hold only for the component designed to capture mispricing."

The Fama and French (F&F) factors do not reliably estimate the size and book-to-market effects. Our paper shows that the former has been underestimated in the US market while the latter overestimated. We do so by replacing F&F's independent rankings by the conditional ones introduced by Lambert and Hubner (2013), over which we improve the sorting procedure. This new specification better reflects the properties of the individual risk premiums. We emphasize a much stronger size effect than conventionally documented. As a major related outcome, the alternative risk factors deliver less specification errors when used to price passive investment indices..

Notable quotations from the academic research paper:

"The paper revisits the size and book-to-market effect in the US market over the 1980-2007 sample period. It demonstrates a strong size but an insignificant book-to-market effect over the sample period. Our result challenges the Fama and French evidence of the presence of a stronger book-to-market than size effect in the US market. Fama and French’s size and book-to-market premiums are indeed shown to be respectively insignificant and positively significant over the analyzed period. Their evidence is partly supported in the standard construction methodology itself as more weight is attributed to the ranking according to the book-to-market dimensions (Fama and French, 1993).

We propose an alternative way to construct the empirical risk factors of Fama and French (1993) that avoids the contamination of the premiums from the correlation structure of the data. Our paper aims indeed at addressing some of the drawbacks identified in this heuristic approach to construct risk factors. Some attention has been drawn to the potential misevaluation of the size and book-to-market effect implied by the way the Fama and French methodology was constructed (Cremers et al., 2010; Huij and Verbeek, 2009; Brooks et al., 2008). The original Fama and French (F&F) method performs a 2x3 sort of US stocks on market capitalization and on book-to-market and forms six two-dimensional portfolios at the intersections of the two independent rankings. The premiums are defined as the spread between the average low- and high-scoring portfolios. Our main argument motivating the modifications brought to the original F&F method is that the independent sorting procedure underlying the formation of the six F&F two-dimensional portfolios distorts the way stocks are ranked into portfolios by placing disproportionate weights between the portfolios.

We follow the methodology of Lambert and Hübner (2013) and apply a generalized Fama and French technique to infer the size, book-to-market and momentum factors from the US stock market over the sample period of 1980-2007. The main innovations of our premiums reside in a monthly rebalancing of the portfolios (underlying the construction of the risk premiums) in order to capture the time-varying dimensions of risk, in a finer size classification and in a conditional sorting of stocks into portfolios. We consider three risk dimensions. The conditional sorting procedure answers the question whether there is still return variation related to the third risk criterion after having controlled for two other risk dimensions. It consists in performing a sequential sort in three stages. The first two sorts are performed on control risks, while we end by the risk dimension to be priced. As in Cremers, Petajusto and Zitzewitz (2010) and in Huij and Verbeek (2009), our paper demonstrates that the book-to-market premium of F&F is overvalued. We perform several asset pricing tests to check the validity and pricing power of our alternative premium specification. Compared to the Fama and French method, our factor construction method better captures the return spread associated with the source of risk to be priced. It maximizes the dispersion in the related source of risk while keeping minimal dispersion in correlated sources of risk. The conditional sorting and the finer size classification contribute to better balance the weights placed on the small/large value/growth portfolios. The great improvement of the new method lies in the reduction of the specification errors when pricing passive benchmark investment portfolios. Besides, without losing in significance, the modified technique is neater and leads to risk premiums that may not necessarily be used jointly in a regression-based model, unlike the original Fama and French factors whose risk exposures are highly sensitive to the inclusion of the other Fama and French risk factors in the regression.

Our paper more generally supports Lambert and Hübner’s (2013) previous evidence that a sequential sorting procedure could be more appropriate to take into consideration the contamination effects between the premiums. We show that the premiums constructed along this way deliver more consistent risk properties while reaching at least the same specification level as the F&F premiums. Given the critical stance of our paper, we have to go quite in depth into the origins of the improvements of the proposed sequential procedure, assorted with various methodological variations, over the original F&F method. The robustness checks deliver clear insights vis-à-vis the key drivers of alternative approach’s pricing performance. It is the replacement of an independent sort by a sequential one that seems to make the largest difference as expected."

I provide a novel risk-based explanation for the profitability of momentum strategies. I show that the past winners and the past losers are differently exposed to the upside and downside market risks. Winners systematically have higher relative downside market betas and lower relative upside market betas than losers. As a result, the winner-minus-loser momentum portfolios are exposed to extra downside market risk, but hedge against the upside market risk. Such asymmetry in the upside and downside risks is a mechanical consequence of rebalancing momentum portfolios. But it is unattractive for an investor because both positive relative downside betas and negative relative upside betas carry positive risk premiums according to the Downside-Risk CAPM. Hence, the high returns to momentum strategies are a mere compensation for their upside and downside risks. The Downside Risk-CAPM is a robust unifying explanation of returns to momentum portfolios, constructed for different geographical and asset markets, and it outperforms alternative multi-factor models.

Notable quotations from the academic research paper:

"I show that the downside risk alone does not fully explain the returns to the cross-section of momentum portfolios because the upside risk plays a significant role too and cannot be neglected. In fact, it is the difference in the downside and upside betas (beta asymmetry) which varies across momentum portfolios the greatest. For any cross-section of momentum portfolios considered, the difference in betas is monotonically increasing from past losers to past winners. As a result, the winner-minus-loser momentum portfolios are exposed to the downside risk, but hedge against the upside risk.

This finding is consistent with a recent study by Daniel and Moskowitz (2014), who show that the winner-minus-loser momentum portfolios tend to crash when the market rebounds after a decline. The momentum crashes occur during the market upturns because these portfolios appear to be long in the low-beta stocks and short in the high-beta stocks picked in the preceding formation period of the declining market. But if the formation period coincides with the growing market, on the contrary, the momentum portfolios appears to be long in the high-beta stocks and short in the low-beta stocks, what leads to their high exposure to the downside risk if the market turns down. Because the momentum portfolios are rebalanced periodically, and because the market changes its trend often, the momentum portfolios appear to have positive downside betas and negative upside betas mechanically. Recent studies by Barroso and Santa-Clara (2015) and Jacobs, Regele and Weber (2015) also show that past winner and loser portfolios have asymmetric return distributions and, as a result, the momentum portfolio returns exhibit significant negative skewness and high kurtosis. Such asymmetry in risks is not attractive for an investor and requires a risk premium.

In the cross-sectional tests, I show that the relative downside beta, which captures the extra downside risk and, hence, the downside-upside risk asymmetry, explains the returns to the momentum portfolios well, whereas the traditional beta has no explanatory power. The relative downside beta premium is approximately 3-4 percent per month, highly statistically significant and similar in magnitude to the estimates obtained for the stock and currency markets (Lettau et al., 2014; Dobrynskaya, 2014).

My findings are similar for all cross-sections of momentum portfolios in different geographical markets and asset classes. I study the US, Global, European, North-American and Asian-Pacific momentum portfolios of individual stocks, global momentum portfolios of country indices, currency momentum portfolios. I show that momentum is a global phenomenon indeed, and its upside-downside risk structure is similar around the world and in different asset markets. I confirm the findings of Asness, Moskowitz, and Pedersen (2013) that momentum strategies in different locations and asset markets share common risks. But the major contribution of this paper is to show that a microfounded theoretical asset-pricing model (namely, the Downside-Risk CAPM – DR-CAPM) previously used to explain stock and currency returns can also explain the momentum returns well."

We contrast the time-series and cross-sectional performance of three popular investment strategies: carry, momentum and value. While considerable research has examined the performance of these strategies in either a directional or cross-asset settings, we offer some insights on the market conditions that favor the application of a particular setting.

Notable quotations from the academic research paper:

"In quantitative cash equity strategies, momentum is almost always traded across assets (relative value) whereas in futures trading, momentum is typically applied directionally. Why? Our goal is to better understand the performance of three popular strategies, carry, momentum and value in different implementations: time-series vs. cross-sectional.

We fill this gap by providing an analysis of both the time-series and cross-section using a broad number of asset classes: equity, fixed income, currencies and commodities. We measure the relative performance of directional vs. cross-asset strategies as well as strategies that combine the information in each dimension. We show that these strategies are largely profitable over our sample - and the best performance is when these strategies are combined.

Contrasting Table 1 Panel A and Table 5 Panel A, we have quite different results for the three styles: value works well in the cross-section, poorly in time-series; carry works about equally well in both cross-section and time-series and momentum works well in time-series, but poorly in the cross-section.

At a basic level, assuming linear signals, cross sectional portfolio weights are equal to time series weights minus the cross sectional average. This average can be thought of as a global factor. Therefore, we can think of a cross sectional portfolio as a time series portfolio hedged for the global factor. Pursuing this line of reasoning, time series momentum will outperform cross sectional momentum to the extent that the global factor is trending. Alternatively, to the extent that the value indicator trades reversion to the mean, time series value investing will do better than cross sectional value investing when the global factor returns are negatively autocorrelated.

So how do we interpret the results from our data exploration? As stated above, momentum outperformance seems to go hand in hand with value underperformance in the time series versus the cross section. Although this result is difficult to interpret, we can offer three possible explanations.

The first is that momentum, unlike value, takes the price movements themselves as being informative and, as in such, may be better placed to assimilate any truly novel information about the global factor which may not be captured by the valuation model. In other words, the momentum model may account unwittingly for the factors omitted by the valuation model. This distinction between value and momentum is most prominent in the more correlated asset classes of equities and bonds.

In FX and commodities where a global factor is much less apparent the performance dierences between the cross section and time series is much less. This makes sense, because in moving from time series to cross sectional portfolios we are essentially hedging out a single global factor. If this factor explains less, there will be less to hedge out and less difference between the portfolios (in either direction). This is exactly what we observe. Although this explanation may have merits, it does not help us understand why value performs better than momentum in the cross section.

Another explanation is that major global factors have exhibited very strong trends which have by definition hurt reversion based value predictors. It may even be claimed that the central purpose of stimulative public policies recently was to boost wealth effects by supporting a sustained rally in stocks and bonds, that in turn favored momentum over value in both asset classes.

A third explanation for time series versus cross sectional performance is the correlations of the signals, and how they compare to the correlations of the underlying markets. All else being equal, a cross sectional approach has more to gain when asset correlations are very high, as the above-mentioned global factor will dominate, and hedging this out will increase diversication by boosting exposure to a wider range of other factors. However, if signals are also highly correlated, a cross sectional approach will hedge out most of the (presumably informative) signals as well, potentially canceling out any gain from the diversication. Conversely if asset correlations are high, but signal
correlations low, we will likely lose very little of the information in the signals by forcing them to be cross-sectional as their information already mostly relates to the non-global factors. This could potentially explain the outperformance of time series momentum against time series value in bonds and equities. Although both asset classes are internally highly correlated, the momentum signals on them are even more correlated, so moving to a cross-sectional framework will potentially hedge out more of the alpha from the signals than noise from the market. Correlations of value signals are notably smaller.

While all of the above explanations have appeal, the readers should note that value is traditionally traded in the cross section while momentum is traded in time series. So it would seem that traders have generally come to the "correct conclusions".

By combining simple signals in carry, momentum and value across less than 100 liquid futures, forwards and swap markets we are able to achieve a remarkably stable strategy over 25 years with a Sharpe ratio of close to 2, returning an approximately eight-fold increase on a hypothetical 15% volatility investment. This can be considered a genuine return, as the strategy has very low funding costs. Is this too good to be true? Why is not every investor trading these styles in combination?

We tried our hand at six possible explanations.

Selection bias is a partial answer. Why did we choose these styles and not others? Because, by and large, they have worked consistently over time and across asset classes. However, in defense of our results, not many styles make sense across such diverse asset classes; so the selection pool is not large.

What about potential over-tting? There was no fitting in this exercise, although some potentially creeps in from experience. Why do our value predictors look back much further than the momentum predictor? Because momentum has worked better at medium frequencies, whereas value is clearly a long-term game. How obvious would this have been 25 years ago?

Survivorship and selection bias of assets is also a problem. Toxic emerging markets may be excluded. This study excluded Argentina but included the likes of Russia, Greece, Indonesia. This kind of bias will likely favor value and carry through the removal of markets where turmoil has caused major assets to exit.

Momentum suffers from another potential bias. Back in 1990, many markets we would include now were much smaller. The ones that make it into our study have likely grown over this time, often via a strong long-term up-trend. By adding data for markets which are now big, but were once small, we likely give a positive bias to momentum predictors.

Another, perhaps more appealing explanation for the performance is simply that few firms have the appetite and patience to trade something so simple. It is easy to forget the arguments in 1999 that value had been replaced by growth, in 2008 that carry was toxic, and in 2011-13 that momentum was finished. These are long-term signals whose performance oscillates over time, with each style experiencing negative performances for at least three years. It is difficult to stick with underperforming strategies this long."